241 research outputs found
Capturing Logarithmic Space and Polynomial Time on Chordal Claw-Free Graphs
We show that the class of chordal claw-free graphs admits LREC-definable
canonization. LREC is a logic that extends first-order logic with counting
by an operator that allows it to formalize a limited form of recursion. This
operator can be evaluated in logarithmic space. It follows that there exists a
logarithmic-space canonization algorithm, and therefore a logarithmic-space
isomorphism test, for the class of chordal claw-free graphs. As a further
consequence, LREC captures logarithmic space on this graph class. Since
LREC is contained in fixed-point logic with counting, we also obtain that
fixed-point logic with counting captures polynomial time on the class of
chordal claw-free graphs.Comment: 34 pages, 13 figure
Trading Determinism for Time in Space Bounded Computations
Savitch showed in that nondeterministic logspace (NL) is contained in
deterministic space but his algorithm requires
quasipolynomial time. The question whether we can have a deterministic
algorithm for every problem in NL that requires polylogarithmic space and
simultaneously runs in polynomial time was left open.
In this paper we give a partial solution to this problem and show that for
every language in NL there exists an unambiguous nondeterministic algorithm
that requires space and simultaneously runs in
polynomial time.Comment: Accepted in MFCS 201
Arithmetic Circuits and the Hadamard Product of Polynomials
Motivated by the Hadamard product of matrices we define the Hadamard product
of multivariate polynomials and study its arithmetic circuit and branching
program complexity. We also give applications and connections to polynomial
identity testing. Our main results are the following. 1. We show that
noncommutative polynomial identity testing for algebraic branching programs
over rationals is complete for the logspace counting class \ceql, and over
fields of characteristic the problem is in \ModpL/\Poly. 2.We show an
exponential lower bound for expressing the Raz-Yehudayoff polynomial as the
Hadamard product of two monotone multilinear polynomials. In contrast the
Permanent can be expressed as the Hadamard product of two monotone multilinear
formulas of quadratic size.Comment: 20 page
The Complexity of Reasoning for Fragments of Autoepistemic Logic
Autoepistemic logic extends propositional logic by the modal operator L. A
formula that is preceded by an L is said to be "believed". The logic was
introduced by Moore 1985 for modeling an ideally rational agent's behavior and
reasoning about his own beliefs. In this paper we analyze all Boolean fragments
of autoepistemic logic with respect to the computational complexity of the
three most common decision problems expansion existence, brave reasoning and
cautious reasoning. As a second contribution we classify the computational
complexity of counting the number of stable expansions of a given knowledge
base. To the best of our knowledge this is the first paper analyzing the
counting problem for autoepistemic logic
Computing Bits of Algebraic Numbers
We initiate the complexity theoretic study of the problem of computing the
bits of (real) algebraic numbers. This extends the work of Yap on computing the
bits of transcendental numbers like \pi, in Logspace.
Our main result is that computing a bit of a fixed real algebraic number is
in C=NC1\subseteq Logspace when the bit position has a verbose (unary)
representation and in the counting hierarchy when it has a succinct (binary)
representation.
Our tools are drawn from elementary analysis and numerical analysis, and
include the Newton-Raphson method. The proof of our main result is entirely
elementary, preferring to use the elementary Liouville's theorem over the much
deeper Roth's theorem for algebraic numbers.
We leave the possibility of proving non-trivial lower bounds for the problem
of computing the bits of an algebraic number given the bit position in binary,
as our main open question. In this direction we show very limited progress by
proving a lower bound for rationals
The Complexity of Bisimulation and Simulation on Finite Systems
In this paper the computational complexity of the (bi)simulation problem over
restricted graph classes is studied. For trees given as pointer structures or
terms the (bi)simulation problem is complete for logarithmic space or NC,
respectively. This solves an open problem from Balc\'azar, Gabarr\'o, and
S\'antha. Furthermore, if only one of the input graphs is required to be a
tree, the bisimulation (simulation) problem is contained in AC (LogCFL). In
contrast, it is also shown that the simulation problem is P-complete already
for graphs of bounded path-width
Longest paths in Planar DAGs in Unambiguous Logspace
We show via two different algorithms that finding the length of the longest
path in planar directed acyclic graph (DAG) is in unambiguous logspace UL, and
also in the complement class co-UL. The result extends to toroidal DAGs as
well
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